Toniolo and Linder, Equation (7)

Average Error: 42.9 → 10.7

Time: 28.6s

Precision: binary64

Cost: 28628

Math

\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

↓

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \frac{2}{x} + \left(2 + \frac{2}{x}\right)\\ t_4 := \frac{\ell \cdot \ell}{x}\\ t_5 := \left(t \cdot t\right) \cdot -2 - \ell \cdot \ell\\ t_6 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ t_7 := \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\\ \mathbf{if}\;t \leq -3.7 \cdot 10^{-5}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-169}:\\ \;\;\;\;\frac{t_2}{\sqrt{t_4 + \left(t_6 + t_1\right)}}\\ \mathbf{elif}\;t \leq -6.8 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(t_4 + t_6\right) + t_7}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-161}:\\ \;\;\;\;\frac{t_2}{\mathsf{fma}\left(t, \sqrt{t_3}, \sqrt{\frac{1}{t_3}} \cdot \left(0.5 \cdot \frac{2 \cdot t_1}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 8800000000000:\\ \;\;\;\;\frac{t_2}{\sqrt{\left(t_4 + \mathsf{fma}\left(-1, \frac{t_5 + t_5}{x \cdot x}, t_6\right)\right) + t_7}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* l (/ l x)))
        (t_2 (* t (sqrt 2.0)))
        (t_3 (+ (/ 2.0 x) (+ 2.0 (/ 2.0 x))))
        (t_4 (/ (* l l) x))
        (t_5 (- (* (* t t) -2.0) (* l l)))
        (t_6 (* 2.0 (+ (* t t) (/ (* t t) x))))
        (t_7 (/ (+ (* l l) (* 2.0 (* t t))) x)))
   (if (<= t -3.7e-5)
     (+ -1.0 (+ (/ 1.0 x) (/ -0.5 (* x x))))
     (if (<= t -3.6e-169)
       (/ t_2 (sqrt (+ t_4 (+ t_6 t_1))))
       (if (<= t -6.8e-235)
         (+ (/ 1.0 x) -1.0)
         (if (<= t 4.6e-301)
           (* (sqrt 2.0) (/ t (sqrt (+ (+ t_4 t_6) t_7))))
           (if (<= t 2.1e-161)
             (/
              t_2
              (fma
               t
               (sqrt t_3)
               (* (sqrt (/ 1.0 t_3)) (* 0.5 (/ (* 2.0 t_1) t)))))
             (if (<= t 8800000000000.0)
               (/
                t_2
                (sqrt (+ (+ t_4 (fma -1.0 (/ (+ t_5 t_5) (* x x)) t_6)) t_7)))
               (sqrt (/ (+ x -1.0) (+ 1.0 x)))))))))))

C

double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

↓

double code(double x, double l, double t) {
	double t_1 = l * (l / x);
	double t_2 = t * sqrt(2.0);
	double t_3 = (2.0 / x) + (2.0 + (2.0 / x));
	double t_4 = (l * l) / x;
	double t_5 = ((t * t) * -2.0) - (l * l);
	double t_6 = 2.0 * ((t * t) + ((t * t) / x));
	double t_7 = ((l * l) + (2.0 * (t * t))) / x;
	double tmp;
	if (t <= -3.7e-5) {
		tmp = -1.0 + ((1.0 / x) + (-0.5 / (x * x)));
	} else if (t <= -3.6e-169) {
		tmp = t_2 / sqrt((t_4 + (t_6 + t_1)));
	} else if (t <= -6.8e-235) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 4.6e-301) {
		tmp = sqrt(2.0) * (t / sqrt(((t_4 + t_6) + t_7)));
	} else if (t <= 2.1e-161) {
		tmp = t_2 / fma(t, sqrt(t_3), (sqrt((1.0 / t_3)) * (0.5 * ((2.0 * t_1) / t))));
	} else if (t <= 8800000000000.0) {
		tmp = t_2 / sqrt(((t_4 + fma(-1.0, ((t_5 + t_5) / (x * x)), t_6)) + t_7));
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return tmp;
}

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

↓

function code(x, l, t)
	t_1 = Float64(l * Float64(l / x))
	t_2 = Float64(t * sqrt(2.0))
	t_3 = Float64(Float64(2.0 / x) + Float64(2.0 + Float64(2.0 / x)))
	t_4 = Float64(Float64(l * l) / x)
	t_5 = Float64(Float64(Float64(t * t) * -2.0) - Float64(l * l))
	t_6 = Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x)))
	t_7 = Float64(Float64(Float64(l * l) + Float64(2.0 * Float64(t * t))) / x)
	tmp = 0.0
	if (t <= -3.7e-5)
		tmp = Float64(-1.0 + Float64(Float64(1.0 / x) + Float64(-0.5 / Float64(x * x))));
	elseif (t <= -3.6e-169)
		tmp = Float64(t_2 / sqrt(Float64(t_4 + Float64(t_6 + t_1))));
	elseif (t <= -6.8e-235)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= 4.6e-301)
		tmp = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(Float64(t_4 + t_6) + t_7))));
	elseif (t <= 2.1e-161)
		tmp = Float64(t_2 / fma(t, sqrt(t_3), Float64(sqrt(Float64(1.0 / t_3)) * Float64(0.5 * Float64(Float64(2.0 * t_1) / t)))));
	elseif (t <= 8800000000000.0)
		tmp = Float64(t_2 / sqrt(Float64(Float64(t_4 + fma(-1.0, Float64(Float64(t_5 + t_5) / Float64(x * x)), t_6)) + t_7)));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t * t), $MachinePrecision] * -2.0), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -3.7e-5], N[(-1.0 + N[(N[(1.0 / x), $MachinePrecision] + N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.6e-169], N[(t$95$2 / N[Sqrt[N[(t$95$4 + N[(t$95$6 + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -6.8e-235], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, 4.6e-301], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(N[(t$95$4 + t$95$6), $MachinePrecision] + t$95$7), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.1e-161], N[(t$95$2 / N[(t * N[Sqrt[t$95$3], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[(N[(2.0 * t$95$1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8800000000000.0], N[(t$95$2 / N[Sqrt[N[(N[(t$95$4 + N[(-1.0 * N[(N[(t$95$5 + t$95$5), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision] + t$95$7), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -3.69999999999999981e-5

   1. Initial program 41.6

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Simplified 41.6

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{x + -1}{\mathsf{fma}\left(t, 2 \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \]
      Proof
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x -1) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (*.f64 2 t)) (*.f64 l l))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 t) t)) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l))))): 7 points increase in error, 19 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))): 3 points increase in error, 12 points decrease in error

   3. Taylor expanded in t around inf 50.2

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \]

   4. Simplified 50.2

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{-1 + x}}}} \]
      Proof
      (*.f64 2 (/.f64 (*.f64 (*.f64 t t) (+.f64 x 1)) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (+.f64 x 1)) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (pow.f64 t 2) (Rewrite=> +-commutative_binary64 (+.f64 1 x))) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 1 x) (pow.f64 t 2))) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (+.f64 1 x) (pow.f64 t 2)) (Rewrite<= +-commutative_binary64 (+.f64 x -1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (+.f64 1 x) (pow.f64 t 2)) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (+.f64 1 x) (pow.f64 t 2)) (Rewrite<= sub-neg_binary64 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in t around -inf 5.2

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]

   6. Simplified 5.2

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]
      Proof
      (neg.f64 (sqrt.f64 (/.f64 (+.f64 x -1) (+.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (sqrt.f64 (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (+.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (sqrt.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (+.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (sqrt.f64 (/.f64 (-.f64 x 1) (+.f64 1 x))))): 0 points increase in error, 0 points decrease in error

   7. Taylor expanded in x around inf 5.5

      \[\leadsto -\color{blue}{\left(\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x}\right)} \]

   8. Simplified 5.5

      \[\leadsto -\color{blue}{\left(1 + \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\right)} \]
      Proof
      (+.f64 1 (-.f64 (/.f64 1/2 (*.f64 x x)) (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (-.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (*.f64 x x)) (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (-.f64 (/.f64 (*.f64 1/2 1) (Rewrite<= unpow2_binary64 (pow.f64 x 2))) (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))) (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 1 (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))) (/.f64 1 x))): 0 points increase in error, 2 points decrease in error

   if -3.69999999999999981e-5 < t < -3.60000000000000001e-169

   1. Initial program 32.1

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Taylor expanded in x around inf 10.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \]

   3. Simplified 10.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 (*.f64 l l) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x))): 0 points increase in error, 1 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (pow.f64 t 2))))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 2 (pow.f64 t 2)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around 0 10.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \]

   5. Simplified 10.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\ell \cdot \frac{-\ell}{x}}\right)}} \]
      Proof
      (*.f64 l (/.f64 (neg.f64 l) x)): 0 points increase in error, 0 points decrease in error
      (*.f64 l (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 l l) x))): 41 points increase in error, 16 points decrease in error
      (neg.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error

   if -3.60000000000000001e-169 < t < -6.79999999999999943e-235

   1. Initial program 63.5

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Simplified 63.5

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{x + -1}{\mathsf{fma}\left(t, 2 \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \]
      Proof
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x -1) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (*.f64 2 t)) (*.f64 l l))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 t) t)) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l))))): 7 points increase in error, 19 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))): 3 points increase in error, 12 points decrease in error

   3. Taylor expanded in t around inf 64.0

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \]

   4. Simplified 64.0

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{-1 + x}}}} \]
      Proof
      (*.f64 2 (/.f64 (*.f64 (*.f64 t t) (+.f64 x 1)) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (+.f64 x 1)) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (pow.f64 t 2) (Rewrite=> +-commutative_binary64 (+.f64 1 x))) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 1 x) (pow.f64 t 2))) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (+.f64 1 x) (pow.f64 t 2)) (Rewrite<= +-commutative_binary64 (+.f64 x -1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (+.f64 1 x) (pow.f64 t 2)) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (+.f64 1 x) (pow.f64 t 2)) (Rewrite<= sub-neg_binary64 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in t around -inf 34.0

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]

   6. Simplified 34.0

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]
      Proof
      (neg.f64 (sqrt.f64 (/.f64 (+.f64 x -1) (+.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (sqrt.f64 (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (+.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (sqrt.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (+.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (sqrt.f64 (/.f64 (-.f64 x 1) (+.f64 1 x))))): 0 points increase in error, 0 points decrease in error

   7. Taylor expanded in x around inf 34.2

      \[\leadsto -\color{blue}{\left(1 - \frac{1}{x}\right)} \]

   if -6.79999999999999943e-235 < t < 4.6000000000000003e-301

   1. Initial program 62.4

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Simplified 62.2

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{x + -1}{\mathsf{fma}\left(t, 2 \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \]
      Proof
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x -1) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (*.f64 2 t)) (*.f64 l l))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 t) t)) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l))))): 7 points increase in error, 19 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))): 3 points increase in error, 12 points decrease in error

   3. Taylor expanded in x around inf 27.3

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \]

   4. Simplified 27.3

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right)\right) - \frac{-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x}}}} \]
      Proof
      (-.f64 (+.f64 (/.f64 (*.f64 l l) x) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t)))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t)))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (*.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t)))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2))))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x)): 0 points increase in error, 1 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (/.f64 (neg.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 2 (*.f64 t t)))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (/.f64 (neg.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   if 4.6000000000000003e-301 < t < 2.1e-161

   1. Initial program 62.6

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Taylor expanded in x around inf 33.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \]

   3. Simplified 33.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 (*.f64 l l) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x))): 0 points increase in error, 1 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (pow.f64 t 2))))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 2 (pow.f64 t 2)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around inf 24.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + 0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right)}} \]

   5. Simplified 24.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(t, \sqrt{\left(2 + \frac{2}{x}\right) + \frac{2}{x}}, \sqrt{\frac{1}{\left(2 + \frac{2}{x}\right) + \frac{2}{x}}} \cdot \left(0.5 \cdot \frac{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}{t}\right)\right)}} \]
      Proof
      (fma.f64 t (sqrt.f64 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x)))) (*.f64 1/2 (/.f64 (*.f64 2 (*.f64 l (/.f64 l x))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (+.f64 2 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)) (/.f64 2 x))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x)))) (*.f64 1/2 (/.f64 (*.f64 2 (*.f64 l (/.f64 l x))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (+.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))) (/.f64 2 x))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x)))) (*.f64 1/2 (/.f64 (*.f64 2 (*.f64 l (/.f64 l x))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 2 1)) (*.f64 2 (/.f64 1 x))) (/.f64 2 x))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x)))) (*.f64 1/2 (/.f64 (*.f64 2 (*.f64 l (/.f64 l x))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 2 (+.f64 1 (/.f64 1 x)))) (/.f64 2 x))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x)))) (*.f64 1/2 (/.f64 (*.f64 2 (*.f64 l (/.f64 l x))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x)))) (*.f64 1/2 (/.f64 (*.f64 2 (*.f64 l (/.f64 l x))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x))))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x)))) (*.f64 1/2 (/.f64 (*.f64 2 (*.f64 l (/.f64 l x))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (+.f64 2 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)) (/.f64 2 x)))) (*.f64 1/2 (/.f64 (*.f64 2 (*.f64 l (/.f64 l x))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (+.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))) (/.f64 2 x)))) (*.f64 1/2 (/.f64 (*.f64 2 (*.f64 l (/.f64 l x))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 2 1)) (*.f64 2 (/.f64 1 x))) (/.f64 2 x)))) (*.f64 1/2 (/.f64 (*.f64 2 (*.f64 l (/.f64 l x))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 2 (+.f64 1 (/.f64 1 x)))) (/.f64 2 x)))) (*.f64 1/2 (/.f64 (*.f64 2 (*.f64 l (/.f64 l x))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)))) (*.f64 1/2 (/.f64 (*.f64 2 (*.f64 l (/.f64 l x))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))))) (*.f64 1/2 (/.f64 (*.f64 2 (*.f64 l (/.f64 l x))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (*.f64 1/2 (/.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (*.f64 l (/.f64 l x))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (*.f64 1/2 (/.f64 (*.f64 (+.f64 1 1) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 l l) x))) t)))): 17 points increase in error, 1 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (*.f64 1/2 (/.f64 (*.f64 (+.f64 1 1) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x)) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (*.f64 1/2 (/.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (/.f64 (pow.f64 l 2) x) (*.f64 1 (/.f64 (pow.f64 l 2) x)))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (*.f64 1/2 (/.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (/.f64 (pow.f64 l 2) x))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (*.f64 1/2 (/.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (/.f64 (pow.f64 l 2) x) (*.f64 -1 (/.f64 (pow.f64 l 2) x)))) t)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 1/2 (/.f64 (-.f64 (/.f64 (pow.f64 l 2) x) (*.f64 -1 (/.f64 (pow.f64 l 2) x))) t)) (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 (/.f64 (-.f64 (/.f64 (pow.f64 l 2) x) (*.f64 -1 (/.f64 (pow.f64 l 2) x))) t) (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (*.f64 1/2 (*.f64 (/.f64 (-.f64 (/.f64 (pow.f64 l 2) x) (*.f64 -1 (/.f64 (pow.f64 l 2) x))) t) (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) t)) (*.f64 1/2 (*.f64 (/.f64 (-.f64 (/.f64 (pow.f64 l 2) x) (*.f64 -1 (/.f64 (pow.f64 l 2) x))) t) (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error

   if 2.1e-161 < t < 8.8e12

   1. Initial program 30.3

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Taylor expanded in x around -inf 10.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \]

   3. Simplified 10.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\frac{\ell \cdot \ell}{x} + \mathsf{fma}\left(-1, \frac{\left(-\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)\right) - \left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)}{x \cdot x}, 2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right)\right)\right) - \frac{-\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)}{x}}}} \]
      Proof
      (-.f64 (+.f64 (/.f64 (*.f64 l l) x) (fma.f64 -1 (/.f64 (-.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (fma.f64 -1 (/.f64 (-.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 -1 (/.f64 (-.f64 (neg.f64 (+.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (*.f64 l l))) (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 -1 (/.f64 (-.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (pow.f64 t 2))))) (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 -1 (/.f64 (-.f64 (neg.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 2 (pow.f64 t 2)))) (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 -1 (/.f64 (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (*.f64 l l))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (pow.f64 t 2))))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 2 (pow.f64 t 2)))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (Rewrite<= unpow2_binary64 (pow.f64 x 2))) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2)) (*.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t))))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2)) (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x)): 0 points increase in error, 1 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2)) (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (*.f64 l l))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))))) (/.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (pow.f64 t 2))))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))))) (/.f64 (neg.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 2 (pow.f64 t 2)))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))))) (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))))) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   if 8.8e12 < t

   1. Initial program 42.3

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Simplified 42.4

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{x + -1}{\mathsf{fma}\left(t, 2 \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \]
      Proof
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x -1) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (*.f64 2 t)) (*.f64 l l))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 t) t)) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l))))): 7 points increase in error, 19 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))): 3 points increase in error, 12 points decrease in error

   3. Taylor expanded in t around inf 52.3

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \]

   4. Simplified 52.3

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{-1 + x}}}} \]
      Proof
      (*.f64 2 (/.f64 (*.f64 (*.f64 t t) (+.f64 x 1)) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (+.f64 x 1)) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (pow.f64 t 2) (Rewrite=> +-commutative_binary64 (+.f64 1 x))) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 1 x) (pow.f64 t 2))) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (+.f64 1 x) (pow.f64 t 2)) (Rewrite<= +-commutative_binary64 (+.f64 x -1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (+.f64 1 x) (pow.f64 t 2)) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (+.f64 1 x) (pow.f64 t 2)) (Rewrite<= sub-neg_binary64 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in t around 0 5.2

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-5}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-169}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -6.8 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-161}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}, \sqrt{\frac{1}{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}} \cdot \left(0.5 \cdot \frac{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 8800000000000:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{\ell \cdot \ell}{x} + \mathsf{fma}\left(-1, \frac{\left(\left(t \cdot t\right) \cdot -2 - \ell \cdot \ell\right) + \left(\left(t \cdot t\right) \cdot -2 - \ell \cdot \ell\right)}{x \cdot x}, 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.7
Cost	23640

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{2 + \frac{4}{x}}\\ t_3 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ t_4 := \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\\ t_5 := \frac{\ell \cdot \ell}{x}\\ t_6 := \left(t \cdot t\right) \cdot -2 - \ell \cdot \ell\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{-6}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-168}:\\ \;\;\;\;\frac{t_1}{\sqrt{t_5 + \left(t_3 + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -4.7 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(t_5 + t_3\right) + t_4}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-161}:\\ \;\;\;\;\frac{t_1}{\frac{\frac{\ell}{t}}{\frac{t_2}{\frac{\ell}{x}}} + t \cdot t_2}\\ \mathbf{elif}\;t \leq 100000000000:\\ \;\;\;\;\frac{t_1}{\sqrt{\left(t_5 + \mathsf{fma}\left(-1, \frac{t_6 + t_6}{x \cdot x}, t_3\right)\right) + t_4}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 2
Error	12.1
Cost	21328

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{2 + \frac{4}{x}}\\ t_3 := \frac{\ell \cdot \ell}{x}\\ t_4 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ \mathbf{if}\;t \leq -3.7 \cdot 10^{-5}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq -6.6 \cdot 10^{-170}:\\ \;\;\;\;\frac{t_1}{\sqrt{t_3 + \left(t_4 + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-234}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(t_3 + t_4\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{\frac{\ell}{t}}{\frac{t_2}{\frac{\ell}{x}}} + t \cdot t_2}\\ \end{array} \]

Alternative 3
Error	10.6
Cost	20304

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -7.9 \cdot 10^{-7}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-168}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-221}:\\ \;\;\;\;\frac{t_1}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-166}:\\ \;\;\;\;\left(1 - \frac{1}{x}\right) + \frac{\frac{0.5}{x}}{x}\\ \mathbf{elif}\;t \leq 33000000000000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 4
Error	11.9
Cost	15440

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ t_3 := \frac{t \cdot \sqrt{2}}{\sqrt{t_1 + \left(t_2 + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -2.95 \cdot 10^{-5}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-169}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-295}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(t_1 + t_2\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-173}:\\ \;\;\;\;\left(1 - \frac{1}{x}\right) + \frac{\frac{0.5}{x}}{x}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 5
Error	11.9
Cost	15320

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \frac{t_2}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + t_1\right)}}\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-169}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -3.15 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-295}:\\ \;\;\;\;\frac{t_2}{\sqrt{2 \cdot t_1}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-167}:\\ \;\;\;\;\left(1 - \frac{1}{x}\right) + \frac{\frac{0.5}{x}}{x}\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{+14}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 6
Error	14.5
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -3.15 \cdot 10^{-235}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-295}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 7
Error	14.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-235}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 6.9 \cdot 10^{-295}:\\ \;\;\;\;\frac{t \cdot \left(-\sqrt{x}\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 8
Error	14.9
Cost	7048

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-235}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-295}:\\ \;\;\;\;\frac{t \cdot \left(-\sqrt{x}\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{x}\right) + \frac{\frac{0.5}{x}}{x}\\ \end{array} \]

Alternative 9
Error	15.7
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{x}\right) + \frac{\frac{0.5}{x}}{x}\\ \end{array} \]

Alternative 10
Error	15.6
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-306}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{x}\right) + \frac{\frac{0.5}{x}}{x}\\ \end{array} \]

Alternative 11
Error	16.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-306}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 12
Error	15.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 13
Error	16.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-306}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	39.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022330 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))